Question

Prove that the straight lines joining the origin to the points of intersection of the straight line x - y = 2 and the curve 5x² + 12xy - 8y² + 8x - 4y + 12 = 0 make equal angles with the axes.

Answer 1

The equation to the required straight lines is

For this equation is homogeneous and therefore represents two straight lines through the origin; also it is satisfied at the points where the two given equations are satisfied.

Now (1) is, on reduction, y² = 4x²

so that the equations to the two lines are

y = 2x and y = -2x.

These lines are equally inclined to the axes.